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On the Origin of Abstraction : Real and Imaginary Parts of Decidability-Making Gilbert Giacomoni To cite this version: Gilbert Giacomoni. On the Origin of Abstraction : Real and Imaginary Parts of Decidability-Making. 11th World Congress of the International Federation of Scholarly Associations of Management (IF- SAM), Jun 2012, Limerick, Ireland. pp.145. hal-00750628 HAL Id: hal-00750628 https://hal-mines-paristech.archives-ouvertes.fr/hal-00750628 Submitted on 11 Nov 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the Origin of Abstraction: Real and Imaginary Parts of Decidability-Making Gilbert Giacomoni Institut de Recherche en Gestion - Université Paris 12 (UPEC) 61 Avenue de Général de Gaulle 94010 Créteil [email protected] Centre de Gestion Scientifique – Chaire TMCI (FIMMM) - Mines ParisTech 60 Boulevard Saint-Michel 75006 Paris [email protected] Abstract: Firms seeking an original standpoint, in strategy or design, need to break with imitation and uniformity. They specifically attempt to understand the cognitive processes by which decision-makers manage to work, individually or collectively, through undecidable situations generated by equivalent possible choices and design innovatively. The behavioral tradition has largely anchored on Simon's early conception of bounded rationality, it is important to engage more explicitly cognitive approaches particularly ones that might link to the issue of identifying novel competitive positions. The purpose of the study is to better understand the regeneration and meta-restructuring processes of knowledge systems triggered by decision makers in order to redefine their decidable space by abstraction. The theoretical breakthroughs liable to account a dual form of reasoning, deductive to prove (then make) equivalence and abstractive to represent (then unmake) it, in subtle mechanisms of decisional symmetry, indiscernibility (antisymmetry) and asymmetry, are presented. A development of a core analytical/conceptual apparatus is proposed as an extension of the most widespread models of rationality based on a real dimension (for preference-making), by adding a visible imaginary one (for abstraction-making) and open up vistas capacity in the fields of information systems, knowledge and decision. This extension takes complex numbers as generalizable objects. Key words: decision-making, equality, indiscernibility, undecidability, imaginary, abstraction, knowledge, information, symmetry-breaking, identity, relation, 2 Real and Imaginary Parts of Decidability-Making “from similarity between elements it is possible to derive [by abstraction] another concept to which no name has yet been given. Instead of “the triangles are similar” we say that “the two triangles are of identical shape” or “the shape of one is identical with that of the other” (Frege, 1883). An original standpoint, in strategy or design, requires breaking imitation and uniformity, that is, symmetry1. Some firms2 seeking to distinguish themselves specifically attempt to understand the cognitive processes by which decision-makers (consumers, managers, etc.) manage to work through undecidable situations generated by equivalent possible choices and design innovatively. A theoretical understanding of the processes by which equivalence is conceived in turn helps explain how its symmetry gets broken. In this respect, it appears relevant to examine the theoretical models liable to account for this dual form of reasoning, deductive (to make an equivalence that is a loss of uniqueness3 and discernibility) and abstractive (to unmake the equivalence recreating a common unique representation). Human thinking and human organizations produce knowledge to design and share representations. To prove the equality of objects from a situation where they are not known as such, a production of knowledge is required that makes a new common unique identity being constructible by abstraction. But from objects already known as equal4, the knowledge produced at the origin to prove their equality is not spontaneously accessible and visible. Only the existing common identity is then obviously imposed on the mind, no other new one being easily constructible. The design of an identity requires a knowledge-based capacity. This is a current problem for design or creation activities. But it is also a problem to design an equality that would not have to be “bounded” in some known contexts only (a perfect universal equality being in fact unachievable and thus at the end undecidable). The behavioral tradition has largely anchored on Simon's early conception of bounded rationality (Simon, 1956, 1969, 1976), it is important to engage more explicitly cognitive approaches particularly ones that might link to the issue of identifying novel competitive positions. The purpose of the study is to better understand the regeneration and meta-restructuring processes of knowledge systems. It presents a development of a core analytical/conceptual apparatus that may potentially open up vistas in the fields of information systems, knowledge and decision. At a methodological level, the research was conducted inductively to trace the theoretical foundations of the most widespread models of rationality, particularly set theory (Cantor, 1883), and pinpoint their limitations (Nagel E. & al; 1989). By tracing those foundations, various fields were explored for approaches5 seeking to complement set theory. The research was conducted deductively to show that those approaches all aim to explain the discernibility properties between equal things, which is inconceivable in set theory and leads to undecidable situations. The conclusion of this deductive phase is to propose a non-contradictory axiomatic system that is more comprehensive6 than set theory and can describe the abstraction processes triggered by decision makers in order to redefine a decidable space by restructuring knowledge spaces. This axiomatic system opens up the possibility to generalize the mathematical formalism of partly real and partly imaginary numbers – notably utilized to describe variational phenomena (spread, etc.) - to the representation of objects. 1 « Symmetry » comes from the Greek sun (together, with), meaning commensurability, proportion, harmony (Apollonius De Perge, 2009) 2 One, in the domain of relief and virtual software computing is a partner since 10 years (in a longitudinal approach) and is looking in particular for new generations of images and virtual simulation sensors. Some others and partners of a research chair are focusing on innovation in the fields of transportation, automobile, software, (etc), and help address the questions behind this research. 3 A=B establishes the existence of two copies of a same object named A or B. 4 Such as ‘These computers are the same…’, ‘It is the same person…’, etc., 5 Fuzzy sets, quasi-sets, concept-knowledge, fuzzy logic, object theory, multidimensional coordinate spaces, abstraction principles, etc. 6 More complete while staying consistent in Gödel’s understanding (Gödel, 1940) 3 If to decide is to choose (note that an intentional non-choice can be considered as a choice7), one can start to examine situations in which choice is impossible, albeit desired. This applies to an undecidable situation as opposed to a decidable situation (considered as such in mathematical logic if there is an algorithm which decides step-by-step between yes or no answers). In mathematical logic, a decision problem involves determining whether a statement is universally valid (Cori & Lascar, 1993). Thus, undecidability appears as the devil of any decision-maker reaching for the heaven of equivalence between possible choices at hand, because, ultimately, there would only be one choice left to make. Contrary to appearances, equivalence appears to be incompatible with choice, as evidenced by any attempt to plan the act of choosing in a box containing equivalent objects. The instruction to choose “any” object does not translate into reality. Either the box is considered as a whole or the objects are differentiated. In the latter case, one way to go is to designate a specific object (for example, the first one if they were ordered beforehand), in other words, introduce probabilities (equiprobabilities, to be specific) which will necessarily result in the selection of one object over the other after the draw (because a random variable is an application between events and distinctive values). In all cases, equivalence will be interrupted so that choice can occur, because this choice is supposedly equivalent to any other. Indifference to equivalent choices may spring from a consumer as for any other decider (a doctor, a pilot, etc). An understanding of the process that leads them to conceive of this equivalence is key as it makes it easier, in turn, to understand the breaking of indifference as a seamlessly “symmetric” vision of possibilities, both individually discernible in space or in time, nameable (A, B, etc) and countable (1, 2, etc) and yet interchangeable
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